Tricky arctans

$\alpha = \arctan(\frac{1}{2})$ , $\beta = \arctan(\frac{1}{3})$ , so $\alpha + \beta = \frac{\pi}{4}$

$\begin{aligned} \arctan \left(\frac 12\right) + \arctan \left(\frac 13\right) = \arctan \left(\frac {\frac 12 + \frac 13}{1-\frac 12 \times \frac 13} \right) = \arctan (1) = \boxed{\dfrac \pi 4} \end{aligned}$ .